{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic. David teaches Chris. (c: Chris; d: David; Txy: x teaches y)\n(A) Tdc (B) Tcd (C) Tcc (D) dTc \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best English interpretation of the given proposition, using the following translation key: Ax: x is an apartment Hx: x is a house Lx: x is large Bxy: x is bigger than y (\u2203x)[(Ax \u2022 Lx) \u2022 (\u2203y)(Hy \u2022 Bxy)]\n(A) Some large houses are bigger than some apartments. (B) Some houses are bigger than all large apartments. (C) Some large apartments are bigger than some houses. (D) Some houses are bigger thatn some large apartments. \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nConstruct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n~G \u2283 H\n~H / G\n(A) Valid (B) Invalid. Counterexample when G and H are true (C) Invalid. Counterexample when G is true and H is false (D) Invalid. Counterexample when H is true and G is false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use the following key to translate the given formula of PL to natural, English sentences.\nA: Marina reads a Percy Jackson book.\nB: Izzy plays Minecraft.\nC: Emily stops working.\nD: Russell makes dinner.\nE: Ashleigh stops by.\n(~B \u2022 E) \u2283 D\n(A) If it's not the case that both Izzy plays Minecraft and Ashleigh stops by then Russell makes dinner. (B) If it's both not the case that Izzy plays Minecraft and Ashleigh stops by then Russell makes dinner. (C) Izzy playing Minecraft and Ashleigh stopping by entails that Russell makes dinner. (D) Russell's making dinner is a sufficient condition for Izzy not playing Minecraft and Ashleigh stopping by. \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best English interpretation of the given proposition, using the following translation key: Ax: x is an apartment Hx: x is a house Lx: x is large Bxy: x is bigger than y (\u2203x)[Hx \u2022 (\u2200y)(Ay \u2283 Bxy)]\n(A) Some houses are smaller than all apartments. (B) Every house is bigger than every apartment. (C) Some apartment is smaller than every house. (D) Some houses are bigger than every apartment. \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nConstruct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n~K \u2261 L / (K \u00b7 L) \u2228 (K \u00b7 ~L)\n(A) Valid (B) Invalid. Counterexample when K and L are true (C) Invalid. Counterexample when K is true and L is false (D) Invalid. Counterexample when L is true and K is false \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the antecedent of the following conditional proposition: If the university does not increase financial aid, either the president fails to approve it or the board of trustees prevents it.\n(A) The university increases financial aid. (B) The university does not increase financial aid. (C) The board of trustees prevents it. (D) The president fails to approve it. \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?\n~E \u2283 ~F\nG \u2283 F\nH \u2228 ~E\nH \u2283 I\n~I\n(A) E \u2283 F (B) F \u2283 G (C) H \u2283 ~E (D) ~H \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nEverybody loves Raymond, or not.\n(A) L (B) ~L (C) L \u2022 ~L (D) L \u2228 ~L \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic. All kings are luckier than all paupers. (Kx: x is a king; Px: x is a pauper; Lxy: x is luckier than y)\n(A) (\u2200x)[Kx \u2283 (\u2200y)(Py \u2283 Lxy)] (B) (\u2200x)[Kx \u2283 (\u2200y)(Lxy \u2283 Py)] (C) (\u2200x)(\u2200y)[(Kx \u2022 Py) \u2283 Lyx] (D) (\u2200x)[(Kx \u2022 (\u2200y)(Py)) \u2283 Lxy] \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following pairs of propositions. Then, using the truth tables, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers.\n(~M \u2283 ~N) \u2228 (O \u2261 N) and (~M \u00b7 N) \u00b7 [(~O \u2228 ~N) \u00b7 (O \u2228 N)]\n(A) Logically equivalent (B) Contradictory (C) Neither logically equivalent nor contradictory, but consistent (D) Inconsistent \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nWhich of the following propositions is an immediate (one-step) consequence in PL of the given premises?\nE \u2283 ~F\n~F \u2283 G\n~G\n(A) E \u2283 F (B) F \u2283 G (C) G \u2283 E (D) ~~F \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nUse indirect truth tables to determine whether the following argument is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nE \u2283 (F \u2228 ~G)\nF \u2283 (~E \u2228 ~G) / ~(E \u2261 F)\n(A) Valid (B) Invalid. Counterexample when E, F, and G are true (C) Invalid. Counterexample when E and F are true and G is false (D) Invalid. Counterexample when E, F, and G are false \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nUse indirect truth tables to determine whether the following argument is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nH \u2261 (~I \u2228 J)\nH \u2228 ~J / ~I\n(A) Valid (B) Invalid. Counterexample when H, I, and J are false (C) Invalid. Counterexample when H and I are true and J is false (D) Invalid. Counterexample when I is true and H and J are false \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic. Some firefighters rescue victims from danger. (Dx: x is danger; Fx: x is a firefighter; Vx: x is a victim; Rxyz: x rescues y from z)\n(A) (\u2203x){Fx \u2022 (\u2200y)[Dy \u2283 (\u2203z)(Vz \u2022 Rxyz)]} (B) (\u2203x){Fx \u2022 (\u2203y)[Vy \u2022 (\u2203z)(Dz \u2022 Rxyz)]} (C) (\u2203x){Fx \u2022 (\u2203y)(\u2203z)[(Vy \u2022 Dz) \u2022 Rxzy]} (D) (\u2200x){Fx \u2283 (\u2203y)[Vy . (\u2203z)(Dz \u2022 Rxyz)]} \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the antecedent of the following conditional proposition: The university raises tuition if, and only if, both the governor approves of it and the board of trustees recommends it.\n(A) The university raises tuition. (B) The governor approves of it. (C) The board of trustees recommends it. (D) None of the above \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic. All planets are smaller than Jupiter. (j: Jupiter; Px: x is a planet; Sxy: x is smaller than y)\n(A) (\u2203x)(Px \u2022 Sxj) (B) (\u2200x)(Px \u2022 Sxj) (C) (\u2200x)(Px \u2283 Sjx) (D) (\u2200x)(Px \u2283 Sxj) \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic. Jos\u00e9 introduces Wilma to Kevin. (j: Jos\u00e9; k: Kevin; w: Wilma; Ixyz: x introduces y to z)\n(A) Iwkj (B) Ijwk (C) Iwjk (D) Ijkw \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic: No artifacts are people.\n(A) ~Pa (B) (\u2200x)(Ax \u2283 ~Px) (C) ~Ap (D) (\u2200x)(Ax \u2022 ~Px) \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nP \u2283 Q\nQ \u00b7 R / ~P \u00b7 R\n(A) Valid (B) Invalid. Counterexample when P, R, and Q are true (C) Invalid. Counterexample when P and Q are true and R is false (D) Invalid. Counterexample when R and Q are true and P is false \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic. Marco moves from Italy to Spain. (i: Italy; m: Marco; s: Spain; Mxyz: x moves to y from z)\n(A) Msim (B) Mmis (C) Mims (D) Mmsi \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best English interpretation of the given proposition, using the following translation key: Ax: x is an apartment Hx: x is a house Lx: x is large Bxy: x is bigger than y (\u2200x){Ax \u2283 (\u2200y)[(Hy \u2022 Ly) \u2283 \u223cBxy]}\n(A) All apartments are not bigger than some large house. (B) No apartment is bigger than any large house. (C) All large houses are not bigger than some apartment. (D) No large house is bigger than all apartments. \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nJ \u2261 ~K\nJ \u2228 L\n~K / L\n(A) Valid (B) Invalid. Counterexample when J and K are true and L is false (C) Invalid. Counterexample when J is true and K and L are false (D) Invalid. Counterexample when K is true and J and L are false \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nPablo will go to the store if, and only if, his brother drives him and pays for gas.\n(A) (P \u2261 D) \u2261 G (B) (D \u2022 G) \u2283 P (C) P \u2283 (D \u2022 G) (D) P \u2261 (D \u2022 G) \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use indirect truth tables to determine whether each set of propositions is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)\nD \u2261 E\nF \u2228 D\n~E \u00b7 ~F\n(A) Inconsistent (B) Consistent. Consistent valuation when D, E, and F are true (C) Consistent. Consistent valuation when D and E are true and F is false (D) Consistent. Consistent valuation when D and F are true and E is false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nThe restaurant serves pie or cheesecake or ice cream for dessert.\n(A) P \u2228 (C \u2228 I) (B) (P \u2228 C) \u2228 I (C) P \u2228 C \u2228 I (D) Options A and B \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nIf Beth rides her bike, she gets to work earlier.\n(A) B \u2283 W (B) W \u2283 B (C) B \u2261 W (D) Both A and B \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic: Either Taylor or Eric is a latte boy.\n(A) Lt \u2228 Le (B) Lx \u2228 Ly (C) tL \u2228 eL (D) Tl \u2283 El \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nUse indirect truth tables to determine whether the following argument is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nE \u2283 F\nG \u2283 ~F / ~G \u2228 ~E\n(A) Valid (B) Invalid. Counterexample when E, F, and G are true (C) Invalid. Counterexample when E and G are true and F is false (D) Invalid. Counterexample when F and G are true and E is false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use indirect truth tables to determine whether each set of propositions is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)\nA \u2283 (~B \u2228 C)\nC \u2283 D\nA \u2228 (B \u00b7 ~D)\nA \u2261 (B \u2283 D)\n(A) Inconsistent (B) Consistent. Consistent valuation when A and D are true and B and C are false (C) Consistent. Consistent valuation when A, B, and D are true and C is false (D) Consistent. Consistent valuation when C and D are true and A and B are false \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nM \u2228 N\n~M \u00b7 O / N\n(A) Valid (B) Invalid. Counterexample when M and O are true and N is false (C) Invalid. Counterexample when M is true and O and N are false (D) Invalid. Counterexample when O is true and M and N are false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?\nU \u2283 W\nW \u2283 (X \u2261 ~Y)\n(X \u2261 ~Y) \u2283 Z\n~Z\n(A) U \u2283 Z (B) U \u2228 W (C) ~W (D) ~(X \u2261 ~Y) \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the antecedent of the following conditional proposition: If the Bees don't win their first game, then both the Aardvarks and the Chipmunks win their first games.\n(A) The Aardvarks win their first game. (B) The Bees don't win their first game. (C) The Chipmunks win their first game. (D) Both the Aardvarks and the Chipmunks win their first games. \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic. Holly loves Frances. (f: Frances; h: Holly; Lxy: x loves y)\n(A) hLf (B) fLh (C) Lfh (D) Lhf \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nUse indirect truth tables to determine whether the following argument is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nP \u2283 (Q \u2228 R)\n(Q \u00b7 S) \u2283 ~P\n~(~Q \u2228 R) / ~P\n(A) Valid (B) Invalid. Counterexample when P, Q, R, and S are true (C) Invalid. Counterexample when P and Q are true and R and S are false (D) Invalid. Counterexample when P, R, and S are true and Q is false \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nT \u2228 U\nW \u00b7 T / U\n(A) Valid (B) Invalid. Counterexample when T and W are true and U is false (C) Invalid. Counterexample when T is true and W and U are false (D) Invalid. Counterexample when W is true and T and U are false \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the antecedent of the following conditional proposition: The university raises tuition if both the governor approves of it and the board of trustees recommends it.\n(A) The university raises tuition. (B) The governor approves of it. (C) The board of trustees recommends it. (D) The governor approves of it and the board of trustees recommends it. \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Identify the antecedent of the following conditional proposition: The restaurant does not have peas, so instead they serve corn and potatoes.\n(A) The restaurant has peas. (B) The restaurant does not have peas. (C) The restaurant serves corn. (D) The restaurant serves potatoes. \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic. All jellybeans are sweeter than all fruit. (Fx: x is a fruit; Jx: x is a jellybean; Sxy: x is sweeter than y)\n(A) (\u2200x)[Fx \u2283 (\u2200y)(Jy \u2283 Sxy)] (B) (\u2200x)(\u2200y)[(Fx \u2022 Jy) \u2022 Sxy] (C) (\u2200x)(\u2200y)[Sxy \u2283 (Fy \u2022 Jx)] (D) (\u2200x)[Jx \u2283 (\u2200y)(Fy \u2283 Sxy)] \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the conclusion of the following argument. That the world is my idea is a truth valid for every living creature, though only man can contemplate it. In doing so, he attains philosophical wisdom. No truth is more absolutely certain than that all that exists for knowledge and therefore this world is only object in relation to subject, perception of a perceiver. The world is an idea. (Arthur Schopenhauer, The World as Will and Idea)\n(A) The world is an idea. (B) That the world is my idea is a truth valid for every living creature, though only man can contemplate it. (C) In doing so, he attains philosophical wisdom. (D) No truth is more absolutely certain than that all that exists for knowledge and therefore this world is only object in relation to subject, perception of a perceiver. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic. Leo is taller than Cathy. (c: Cathy; l: Leo; Txy: x is taller than y)\n(A) Tlc (B) Tcl (C) lTc (D) cTl \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following pairs of propositions. Then, using the truth tables, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers.\nC \u2283 (D \u2228 C) and C \u00b7 ~D\n(A) Logically equivalent (B) Contradictory (C) Neither logically equivalent nor contradictory, but consistent (D) Inconsistent \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following pairs of propositions. Then, using the truth tables, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers.\n~O \u2283 P and O \u2228 P\n(A) Logically equivalent (B) Contradictory (C) Neither logically equivalent nor contradictory, but consistent (D) Inconsistent \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nUse indirect truth tables to determine whether the following argument is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n(T \u00b7 ~U) \u2283 ~W\n~W \u2283 X\n~Z \u2283 W\n~(X \u00b7 Z) / T \u2283 U\n(A) Valid (B) Invalid. Counterexample when T, X, and Z are true and U and W are false (C) Invalid. Counterexample when T and X are true and U, W, and Z are false (D) Invalid. Counterexample when T, W, and Z are true and U and X are false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic: Cecilia breaks my heart if, and only if, Maria is either a problem or a girl.\n(A) Bc \u2261 (Pm \u2228 Gm) (B) Bc \u2261 (Pm \u2022 Gm) (C) Bc \u2283 (Pm \u2228 Gm) (D) Cb \u2283 (Pm \u2022 Gm) \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic: If Delilah is in New York City, then Juana is in Bogota or Layla is in San Juan.\n(A) Nd \u2283 (Bj \u2022 Sl) (B) Dn \u2261 (Bx \u2022 Ly) (C) Nd \u2261 (Bj \u2022 Sl) (D) Nd \u2283 (Bj \u2228 Sl) \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use the following key to translate the given formula of PL to natural, English sentences.\nA: Marina reads a Percy Jackson book.\nB: Izzy plays Minecraft.\nC: Emily stops working.\nD: Russell makes dinner.\nE: Ashleigh stops by.\n~(A \u2283 B) \u2022 (B \u2283 ~E)\n(A) It's not the case that Marina's reading a Percy Jackson book entails that Izzy plays Minecraft, but Izzy's playing Minecraft does entail that Ashleigh doesn't stop by. (B) If Marina doesn't read a Percy Jackson book, then Izzy plays Minecraft, which entails that Ashleigh doesn't stop by. (C) Marina's reading a Percy Jackson book does not entail that Izzy plays Minecraft, but Izzy plays Minecraft provided that Ashleigh doesn't stop by. (D) It's not true that Marina reads a Percy Jackson book only when Izzy plays Minecraft, but Izzy plays Minecraft only when Ashleigh stops by. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nUse indirect truth tables to determine whether the following argument is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n(A \u2228 B) \u2283 C\nC \u2283 ~D / D \u2283 A\n(A) Valid (B) Invalid. Counterexample when B, C, and D are true and A is false (C) Invalid. Counterexample when B and D are true and A and C are false (D) Invalid. Counterexample when A, B, and C are false and D is true \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best English interpretation of the given arguments in predicate logic.\nDh \u2283 ~Pt\n(\u2200x)Px \u2228 (\u2200x)Mx\n~Mb\t/ ~Dh\n(A) If my headache is dualist state, then your tickle is a physical state. Either everything is physical or everything is mental. But my broken toe is not a mental state. So my headache is not a dualist state. (B) If my headache is dualist state, then your tickle is not a physical state. Either everything is physical or everything is mental. But my broken toe is not a mental state. So my headache is not a dualist state. (C) If my headache is dualist state, then your tickle is not a physical state. If everything is physical then everything is mental. But my broken toe is not a mental state. So my headache is not a dualist state. (D) If my headache is dualist state, then your tickle is not a physical state. Everything is either physical or mental. But my broken toe is not a mental state. So my headache is not a dualist state. \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?\nA \u2283 (B \u2022 C)\n~(B \u2022 C)\nA \u2228 (D \u2228 B)\n(A) A (B) ~A (C) B (D) ~B \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the following propositions is an immediate (one-step) consequence in PL of the given premises?\n(N \u2283 ~O) \u2283 (O \u2228 Q)\nP \u2283 (N \u2283 ~O)\nN \u2283 P\n(A) P (B) N \u2283 ~O (C) O \u2228 Q (D) ~P \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic. Earth isn't bigger than Jupiter. (e: Earth; j: Jupiter; Bxy: x is bigger than y)\n(A) ~Bje (B) Bej (C) Bje~ (D) ~Bej \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the antecedent of the following conditional proposition: The Bees winning their first game is necessary for either the Aardvarks or the Chipmunks not winning their first game.\n(A) The Aardvarks do not win their first game. (B) The Bees win their first game. (C) The Chipmunks do not win their first game. (D) Either the Aardvarks or the Chipmunks do not win their first game. \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the antecedent of the following conditional proposition: The Bees win their first game only if either the Aardvarks or the Chipmunks do not win their first games.\n(A) The Aardvarks do not win their first game. (B) The Bees win their first game. (C) The Chipmunks do not win their first game. (D) Either the Aardvarks or the Chipmunks do not win their first games. \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic: Some athletes don't get injured.\n(A) ~Ia (B) (\u2203x)(Ax \u2022 ~Ix) (C) ~Ai (D) (\u2203x)(Ix \u2228 ~Ax) \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following pairs of propositions. Then, using the truth tables, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers.\nI \u2261 ~H and ~(~I \u2261 H)\n(A) Logically equivalent (B) Contradictory (C) Neither logically equivalent nor contradictory, but consistent (D) Inconsistent \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use indirect truth tables to determine whether each set of propositions is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)\n(T \u2283 U) \u2283 (W \u2283 U)\nT \u2283 ~(U \u2283 X)\n(T \u2283 ~X) \u2283 W\n(A) Inconsistent (B) Consistent. Consistent valuation when U, W, and X are true and T is false (C) Consistent. Consistent valuation when W and X are true and T and U are false (D) Consistent. Consistent valuation when T and U are true and W and X are false \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic: Caroline is sweet if, and only if, Janet is engaged to Brad.\n(A) Cs \u2283 Ej (B) Sc \u2261 Ej (C) sC \u2261 eJ (D) Sx \u2261 Jy \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nEither England's importing beef is a sufficient condition for France's subsidizing agriculture, or China doesn't promote human rights when and only when South Africa supplies diamonds.\n(A) (E \u2283 F) \u2228 (~C \u2283 S) (B) (F \u2283 E) \u2228 (S \u2283 ~C (C) (E \u2283 F) \u2228 ~(S \u2261 C) (D) (E \u2283 F) \u2228 (~C \u2261 S) \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use the following key to translate the given formula of PL to natural, English sentences.\nA: Marina reads a Percy Jackson book.\nB: Izzy plays Minecraft.\nC: Emily stops working.\nD: Russell makes dinner.\nE: Ashleigh stops by.\n(~A \u2228 ~B) \u2283 C\n(A) If neither Marina reads a Percy Jackson book nor Izzy plays Minecraft, then Emily stops working. (B) If it's not the case that both Marina reads a Percy Jackson book and Izzy plays Minecraft, then Emily stops working. (C) Either Marina not reading a Percy Jackson book or Izzy not playing Minecraft entails that Emily stops working. (D) Emily stopping working is a sufficient condition for either Marina not reading a Percy Jackson book or Izzy not playing Minecraft. \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Identify the conclusion of the following argument. There are those who use predicate variables; the values of these variables are attributes. Predicates are wanted in all sentences regardless of whether there are attributes to refer to. The dummy predicate 'F' is wanted generally for expository purposes. If we are going to quantify over attributes and refer to them, then clarity is served by using recognizable variables and distinctive names for the purposes, and not mixing these up with the predicates. (Quine, Philosophy of Logic)\n(A) There are those who use predicate variables; the values of these variables are attributes. (B) Predicates are wanted in all sentences regardless of whether there are attributes to refer to. (C) The dummy predicate 'F' is wanted generally for expository purposes. (D) If we are going to quantify over attributes and refer to them, then clarity is served by using recognizable variables and distinctive names for the purposes, and not mixing these up with the predicates. \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use the following key to translate the given formula of PL to natural, English sentences.\nA: Marina reads a Percy Jackson book.\nB: Izzy plays Minecraft.\nC: Emily stops working.\nD: Russell makes dinner.\nE: Ashleigh stops by.\n~C \u2228 D\n(A) If Emily doesn't stop working then Russell makes dinner. (B) Emily stops working unless Russell makes dinner. (C) Emily stops working unless Russell doesn't make dinner. (D) Emily doesn't stop working unless Russell makes dinner. \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic: Abdul and Cleopatra are Egyptian.\n(A) Ea \u2022 Ec (B) Ea \u2022 c (C) Ae \u2228 Ce (D) Ex \u2022 Ey \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic: Sheena is a punk rocker.\n(A) Sx (B) xS (C) sP (D) Ps \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic. Kevin is introduced to Jos\u00e9 by Wilma. (j: Jos\u00e9; k: Kevin; w: Wilma; Ixyz: x introduces y to z)\n(A) Iwjk (B) Ijkw (C) Ikjw (D) Iwkj \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic: If Eileen plays fiddle then Sherri sings.\n(A) Fe \u2228 Ss (B) eF \u2283 Ss (C) Fe \u2228 Es (D) Fe \u2283 Ss \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nConstruct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n~C \u2283 D\nD \u2283 C / C\n(A) Valid (B) Invalid. Counterexample when C and D are true (C) Invalid. Counterexample when C is true and D is false (D) Invalid. Counterexample when D is true and C is false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic:: Some blue herons migrate.\n(A) (\u2200x)[(Hx \u2022 Bx) \u2022 Mx] (B) (\u2203x)[(Hx \u2022 Bx) \u2283 Mx] (C) (\u2200x)[(Hx \u2022 Bx) \u2283 Mx] (D) (\u2203x)[(Hx \u2022 Bx) \u2022 Mx] \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best English interpretation of the given arguments in predicate logic.\n(\u2203x)(Cx \u2022 Ox)\n(\u2200x)[(~Cx \u2283 ~Bx) \u2283 ~Og]\t/ ~Og\n(A) Some cookies have oatmeal. If something's not being a cookie entails that it doesn't have chocolate chips, then this cookie doesn't have oatmeal. So this cookie doesn't have oatmeal. (B) Some cookies have oatmeal. If something is not a cookie and does not have chocolate chips, it doesn't have oatmeal. So this cookie doesn't have oatmeal. (C) Some cookies have oatmeal. If something doesn't have oatmeal, then it is not a cookie and it doesn't have chocolate chips. So this cookie doesn't have oatmeal. (D) Some cookies have oatmeal. If something's not having chocolate chips entails that it is not a cookie, then it doesn't have oatmeal. So this cookie doesn't have oatmeal. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the conclusion of the following argument. All of psychology has gotten stuck in moral prejudices and fears. No one has come close to understanding it as the development of the will to power. However, if a person even begins to regard the effects of hatred, envy, covetousness, and the lust to rule as conditions of life and furthermore, as factors essential to the general economy of life, he will begin to get seasick. At this point, he begins to lose himself, and sail over morality. Thus, psychology becomes again the path to fundamental problems. (Friedrich Nietzsche, Beyond Good and Evil)\n(A) All of psychology has gotten stuck in moral prejudices and fears. (B) No one has come close to understanding it as the development of the will to power. (C) If a person even begins to regard the effects of hatred, envy, covetousness, and the lust to rule as conditions of life and furthermore, as factors essential to the general economy of life, he will begin to get seasick. (D) Psychology becomes again the path to fundamental problems. \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use indirect truth tables to determine whether each set of propositions is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)\n~N \u2261 (O \u00b7 P)\n~O \u2283 Q\nN \u00b7 ~Q\n(A) Inconsistent (B) Consistent. Consistent valuation when N, O, and P are true and Q is false (C) Consistent. Consistent valuation when N and O are true and P and Q are false (D) Consistent. Consistent valuation when N and P are true and O and Q are false \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nA person is considered to have lived with you for all of the present year if the person was born or died in the present year and your home was this person's home for the entire time this person was alive.\n(A) [(B \u2228 D) \u2022 H] \u2283 L (B) [L \u2283 (B \u2228 D)] \u2022 H (C) L \u2283 [(B \u2228 D) \u2022 H] (D) (B \u2228 D \u2022 H) \u2283 L \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nConstruct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n(O \u2261 P) \u2228 P / P \u2228 ~O\n(A) Valid (B) Invalid. Counterexample when O and P are true (C) Invalid. Counterexample when O is true and P is false (D) Invalid. Counterexample when P is true and O is false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic: Leroy is bad only if DJ Khaled shouts his name.\n(A) Sk \u2261 Bl (B) Bl \u2261 Sk (C) Bx \u2283 Sy (D) Bl \u2283 Sk \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nIf either George enrolls or Harry enrolls, then Ira does not enroll.\n(A) (G \u2283 H) \u2228 I (B) (G \u2228 H) \u2283 I (C) (G \u2283 H) \u2228 ~I (D) (G \u2228 H) \u2283 ~I \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic: Ms. Jackson lent me some sugar, but neither Janet nor Latoya did.\n(A) Lm \u2228 ~(Jl \u2228 Ll) (B) Lm \u2022 ~(Lj \u2022 Ll) (C) Lm \u2022 ~(Lj \u2228 Ll) (D) Lm \u2228 ~(Lj \u2022 Ll) \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic. Some animals are neglected by cruel people. (Ax: x is an animal; Cx: x is cruel; Px: x is a person; Nxy: x neglects y)\n(A) (\u2203x)[(Ax \u2022 Cx) \u2022 (\u2203y)(Py \u2022 Nyx)] (B) (\u2203x)[(Cx \u2022 Px) \u2022 (\u2203y)(Ay \u2022 Nxy)] (C) (\u2203x)[(Ax \u2022 Px) \u2022 (\u2203y)(Cy \u2022 Nxy)] (D) (\u2203x)(\u2203y){[(Ax \u2022 Cy) \u2022 Py] \u2022 Nxy)]} \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use indirect truth tables to determine whether each set of propositions is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.)\n(E \u2283 F) \u2283 G\n(~E \u2228 F) \u2261 (H \u2228 I)\nI \u2283 (J \u2283 ~H)\n~G \u2283 ~J\n(A) Inconsistent (B) Consistent. Consistent valuation when E is true and F, G, H, I, and J are false (C) Consistent. Consistent valuation when E and H are true and F, G, I, and J are false (D) Consistent. Consistent valuation when E and I are true and F, G, H, and J are false \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic: Some students work hard.\n(A) (\u2203x)(Wx \u2228 Sx) (B) (\u2203x)(Sx \u2228 Wx) (C) (\u2203x)(Sx \u2022 Wx) (D) (\u2203x)(Sx \u2283 Wx) \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nEither England's importing beef is not a necessary condition for France's subsidizing agriculture, or China promotes human rights if South Africa supplies diamonds.\n(A) (~F \u2283 E) \u2228 (C \u2261 ~S) (B) (~E \u2283 F) \u2228 (C \u2283 S) (C) (~F \u2283 E) \u2228 (C \u2261 S) (D) ~(F \u2283 E) \u2228 (S \u2283 C) \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n~A \u00b7 ~B\n(A \u2228 C) \u2228 B / C\n(A) Valid (B) Invalid. Counterexample when A and B are true and C is false (C) Invalid. Counterexample when A is true and B and C are false (D) Invalid. Counterexample when B is true and A and C are false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic. All birds live in some nest. (Bx: x is a bird; Nx: x is a nest; Lxy: x lives in y)\n(A) (\u2200x)(\u2200y)[(Bx \u2022 Ny) \u2283 Lxy] (B) (\u2200x)[Bx \u2283 (\u2203y)(Ny \u2022 Lxy)] (C) (\u2200x)[(\u2203y)(Ny \u2022 Bx) \u2283 Lxy] (D) (\u2200x)(\u2203y)[(Bx \u2022 Ny) \u2283 Lxy] \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the conclusion of the following argument. Utilitarian theory is embarrassed by the possibility of utility monsters. A utility monster gets enormous gains in utility, greater than the utility lost by the sacrifices of others. The sacrifices of even large numbers of other people are outweighed by the gains of the utility monster. The theory seems to require that we all be sacrificed in the monste's maw in order to increase total utility. (Robert Nozick, Anarchy, State, and Utopia)\n(A) Utilitarian theory is embarrassed by the possibility of utility monsters. (B) A utility monster gets enormous gains in utility. (C) A utility monster gets gains in utility greater than the utility lost by the sacrifices of others. (D) The sacrifices of even large numbers of other people are outweighed by the gains of the utility monster. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use the following key to translate the given formula of PL to natural, English sentences.\nA: Marina reads a Percy Jackson book.\nB: Izzy plays Minecraft.\nC: Emily stops working.\nD: Russell makes dinner.\nE: Ashleigh stops by.\n(A \u2228 ~E) \u2261 (D \u2022 ~B)\n(A) Either Marina reading a Percy Jackson book or Ashleigh not stopping by is a necessary condition for both Russell making dinner and Izzy not playing Minecraft. (B) Either Marina reading a Percy Jackson book or Ashleigh not stopping by is a necessary and sufficient condition for Russell making dinner and Izzy playing Minecraft. (C) Marina reads a Percy Jackson book unless Ashleigh stops by if, and only if, both Russell makes dinner and Izzy doesn't play Minecraft. (D) Either Marina reads a Percy Jackson book or Ashleigh does not stop by just in case both Russell makes dinner and Izzy doesn't play Minecraft. \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nBrittany likes fish and lizards, but not cats.\n(A) F \u2022 L \u2022 ~C (B) (F \u2022 L) \u2022 ~C (C) (F \u2022 L) \u2228 ~C (D) C \u2022 (F \u2022 L) \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nIf either Harry enrolls or George does not enroll, then Jim enrolls.\n(A) H \u2228 (~G \u2283 J) (B) ~(H \u2228 G) \u2283 J (C) (H \u2228 ~G) \u2283 J (D) (H \u2283 ~G) \u2228 J \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n~X \u2283 Y\nY \u2283 Z\n~Z / ~X\n(A) Valid (B) Invalid. Counterexample when X, Y, and Z are true (C) Invalid. Counterexample when Y and Z are true and Z is false (D) Invalid. Counterexample when X is true and Y and Z are false \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n S Select the best translation into predicate logic. Some kind people help some endangered species. (Ex: x is endangered; Kx: x is kind; Px: x is a person; Sx: x is a species; Hxy: x helps y)\n(A) (\u2203x){(Kx \u2022 Px) \u2022 (\u2203y)[(Ey \u2022 Sy) \u2022 Hxy]} (B) (\u2203x)(\u2203y){[(Kx \u2022 Sx) \u2022 (Ey \u2022 Py)] \u2022 Hxy} (C) (\u2203x)(\u2203y){[(Kx \u2022 Sx) \u2022 (Ey \u2022 Py)] \u2022 Hyx} (D) (\u2203x)(\u2203y){[(Kx \u2022 Px) \u2022 (Ey \u2022 Sy)] \u2283 Hxy} \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best English interpretation of the given proposition, using the following translation key: Ax: x is an apartment Hx: x is a house Lx: x is large Bxy: x is bigger than y (\u2200x)[Ax \u2283 (\u2203y)(Hy \u2022 Bxy)]\n(A) Every apartment is bigger than some house. (B) Every house is bigger than all apartments. (C) Every house is bigger than some apartment. (D) Some house is bigger than every apartment. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nQ \u2261 R\n~(S \u2228 Q) / R\n(A) Valid (B) Invalid. Counterexample when Q and S are true and R is false (C) Invalid. Counterexample when Q is true and S and R are false (D) Invalid. Counterexample when Q, S, and R are false \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the antecedent of the following conditional proposition: The university raises tuition only if both the governor approves of it and the board of trustees recommends it.\n(A) The university raises tuition. (B) The governor approves of it. (C) The board of trustees recommends it. (D) The governor approves of it and the board of trustees recommends it. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nWhich of the following propositions is an immediate (one-step) consequence in PL of the given premises?\n(A \u2228 B) \u2283 (C \u2228 D)\nA \u2228 B\n~C\n(A) ~A \u2228 ~B (B) ~~C (C) C \u2228 D (D) (A \u2228 B) \u2283 D \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the antecedent of the following conditional proposition: When the university raises tuition, then either the governor approves of it or the board of trustees doesn't prevent it.\n(A) The university raises tuition. (B) The governor approves of it. (C) The board of trustees prevents it. (D) The board of trustees doesn't prevent it. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic. Marco doesn't move from Spain to Italy. (i: Italy; m: Marco; s: Spain; Mxyz: x moves to y from z)\n(A) ~Mmsi (B) M~mis (C) ~Mmis (D) M~msi \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n~S \u2228 T\n~S \u00b7 U\n~T \u2228 U / T \u00b7 U\n(A) Valid (B) Invalid. Counterexample when S, T, and U are true (C) Invalid. Counterexample when S and U are true and T is false (D) Invalid. Counterexample when U is true and S and T are false \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nDoctors are helpful unless their patients are rude.\n(A) H \u2261 R (B) H \u2022 R (C) H \u2228 ~R (D) H \u2228 R \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Use indirect truth tables to determine whether the following argument is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nL \u2283 [(M \u2228 ~N) \u2283 O]\n(N \u2283 O) \u2283 (~P \u2283 Q)\nR \u2283 ~Q / L \u2283 (R \u2283 P)\n(A) Valid (B) Invalid. Counterexample when L, M, O, Q, and R are true and N and P are false (C) Invalid. Counterexample when L, N, O, Q, and R are true and M and P are false (D) Invalid. Counterexample when L, N, and R are true and M, O, P, and Q are false \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the conclusion of the following argument. The sense of a term is supposed to determine its reference. But reference can vary without any difference in thought. So, the senses of terms must be able to vary without variation in thought. So, our thoughts do not determine the meanings of our terms; meanings are not in the head. (Hilary Putnam)\n(A) The sense of a term is supposed to determine its reference. (B) But reference can vary without any difference in thought. (C) Reference can vary without any difference in thought. (D) Our thoughts do not determine the meanings of our terms; meanings are not in the head. \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following pairs of propositions. Then, using the truth tables, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers.\n(G \u2228 ~H) \u2283 G and ~G \u2261 (~H \u00b7 G)\n(A) Logically equivalent (B) Contradictory (C) Neither logically equivalent nor contradictory, but consistent (D) Inconsistent \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nEither England's importing beef is a necessary condition for France's subsidizing agriculture or China's promoting human rights is not a sufficient condition for South Africa's supplying diamonds.\n(A) (E \u2261 F) \u2228 ~(C \u2261 S) (B) (E \u2261 F) \u2228 (~C \u2283 S) (C) (E \u2283 F) \u2228 ~(C \u2283 S) (D) (F \u2283 E) \u2228 ~(C \u2283 S) \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best English interpretation of the given arguments in predicate logic.\nWn \u2228 Wm\n(\u2200x)[Lx \u2283 (Dx \u2283 ~Wx)]\nLn \u2022 Dn\t/ ~(\u2200x)~Wx\n(A) Either Nancy or Marvin are at work. All lawyers are not at work if they are out to dinner. Nancy is a lawyer and out to dinner. So not everything is not at work. (B) Either Nancy or Marvin are at work. All lawyers are out to dinner if they are not at work. Nancy is a lawyer and out to dinner. So not everything is not at work. (C) Either Nancy or Marvin are at work. All lawyers are out to dinner if they are not at work. Nancy is a lawyer and out to dinner. So not everything is at work. (D) Either Nancy or Marvin are at work. All lawyers are not at work if they are out to dinner. Nancy is a lawyer and out to dinner. So not everything is at work. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the conclusion of the following argument. Remember to remove any conclusion indicators. It is wrong for society to kill a murderer. This follows for the reason that if a murderer is wrong in killing his victim, then society is also wrong in killing the murderer. And a murderer is wrong in killing his victim.\n(A) It is wrong for society to kill a murderer. (B) This follows for the reason that if a murderer is wrong in killing his victim, then society is also wrong in killing the murderer. (C) If a murderer is wrong in killing his victim, then society is also wrong in killing the murderer. (D) And a murderer is wrong in killing his victim. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the following propositions is not an immediate (one-step) consequence in PL of the given premises?\nI \u2283 (J \u2261 K)\n(J \u2261 K) \u2283 (L \u2228 M)\nI \u2283 ~L\n(A) ~L (B) J \u2261 K (C) I \u2283 (L \u2228 M) (D) ~L \u2283 (J \u2261 K) \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nConstruct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nI \u2261 ~J\nI \u2228 J / I\n(A) Valid (B) Invalid. Counterexample when I and J are true (C) Invalid. Counterexample when I is true and J is false (D) Invalid. Counterexample when J is true and I is false \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic: Some folks are lucky.\n(A) (\u2203x)(Fx \u2022 Lx) (B) Fl (C) Lf (D) (\u2200x)(Fx \u2022 Lx) \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic. Alexis is greeted by Ben. (a: Alexis; b: Ben; Gxy: x greets y)\n(A) Gab (B) ~Gba (C) aGb (D) Gba \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic: Not all lobsters are red.\n(A) ~(\u2200x)(Lx \u2283 Rx) (B) ~(\u2203x)(Lx \u2283 Rx) (C) ~(\u2203x)(Lx \u2022 Rx) (D) ~(\u2200x)(Lx \u2022 Rx) \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nUse indirect truth tables to determine whether the following argument is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\nH \u2261 (I \u00b7 ~J)\nK \u2261 ~H\n~(K \u2283 J) / I \u2261 J\n(A) Valid (B) Invalid. Counterexample when I, H, and K are true and J is false (C) Invalid. Counterexample when I and H are true and J and K are false (D) Invalid. Counterexample when K is true and I, H, and J are false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following pairs of propositions. Then, using the truth tables, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers.\n~(J \u2228 K) \u00b7 L and (L \u2283 J) \u00b7 K\n(A) Logically equivalent (B) Contradictory (C) Neither logically equivalent nor contradictory, but consistent (D) Inconsistent \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic: All first-year students take a swim test.\n(A) (\u2200x)(Sx \u2283 Fx) (B) (\u2200x)(Fx \u2283 Sx) (C) (\u2200x)(Fx \u2228 Sx) (D) (\u2200x)(Fx \u2261 Sx) \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nSelect the best translation into predicate logic: Johnny came late.\n(A) jL (B) Lj (C) Cj \u2228 Lj (D) Lx \nA: Let's think step by step.", "label": "(B)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic: No senior graduates without a major.\n(A) (\u2200x)(Sx \u2283 ~Gx) (B) ~(\u2200x)(Sx \u2283 Gx) (C) (\u2200x)( ~Sx \u2283 ~Gx) (D) (\u2200x) ~(Sx \u2283 ~Gx) \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the conclusion of the following argument. From a historical point of view, genetics is a young science. The principles that govern inheritance traits were described less than 150 years ago. Furthermore, around the turn of the twentieth century, the laws of inheritance were rediscovered. But even then, the importance of DNA was not really understood until the 1950s.\n(A) But even then, the importance of DNA was not really understood until the 1950s. (B) The importance of DNA was not really understood until the 1950s. (C) From a historical point of view, genetics is a young science. (D) The principles that govern inheritance traits were described less than 150 years ago. \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nEither England's importing beef is a necessary condition for France's subsidizing agriculture or China promotes human rights given that South Africa doesn't supply diamonds.\n(A) (E \u2283 F) \u2228 (C \u2261 ~S) (B) (F \u2283 E) \u2228 (~S \u2261 C) (C) (E \u2283 F) \u2228 (S \u2283~C) (D) (F \u2283 E) \u2228 (~S \u2283 C) \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nEither England's importing beef is a sufficient condition for France's subsidizing agriculture or China's promoting human rights is a necessary condition for South Africa's supplying diamonds.\n(A) (E \u2283 F) \u2228 (S \u2283 C) (B) (F \u2283 E) \u2228 (C \u2283 S) (C) (E \u2283 F) \u2228 (C \u2283 S) (D) (F \u2283 E) \u2228 (S \u2283 C) \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nDylan likes neither math nor science.\n(A) M \u2228 S (B) ~(M \u2022 S) (C) ~(M \u2228 S) (D) ~M \u2228 ~S \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic: All robots are artifacts.\n(A) (\u2200x)(Ax \u2228 Rx) (B) (\u2200x)(Ax \u2022 Rx) (C) (\u2200x)(Rx \u2283 Ax) (D) (\u2200x)(Rx \u2022 Ax) \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nSince the university raised tuition, the president approved of it and the board didn't prevent it.\n(A) (P \u2022 ~B) \u2283 U (B) (U \u2283 P) \u2022 B (C) U \u2283 (P \u2022 ~B) (D) U \u2283 (P \u2228 ~B) \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nIdentify the conclusion of the following argument. We shouldn't abolish capital punishment. If we do, prisons will become more crowded. If prisons become more crowded, then we will have to build more prisons. And we don't want to build more prisons.\n(A) We shouldn't abolish capital punishment. (B) If we do, prisons will become more crowded. (C) If prisons become more crowded, then we will have to build more prisons. (D) We will have to build more prisons. \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Which of the given formulas of PL is the best symbolization of the following sentence?\nEither England's not importing beef is a necessary condition for France's subsidizing agriculture, or China promotes human rights if and only if South Africa doesn't supply diamonds.\n(A) (F \u2283 ~E) \u2228 (C \u2261 ~S) (B) (~E \u2283 F) \u2228 (C \u2283 ~S) (C) ~(F \u2283 E) \u2228 (~S \u2283 C) (D) (F \u2283 E) \u2228 (C \u2261 ~S) \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nUse indirect truth tables to determine whether the following argument is valid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n(K \u00b7 L) \u2283 M\nN \u2283 ~M\nN \u2228 K / L \u2283 K\n(A) Valid (B) Invalid. Counterexample when L, M, and N are true and K is false (C) Invalid. Counterexample when L and M are true and K and N are false (D) Invalid. Counterexample when L and N are true and K and M are false \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Construct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n(G \u2261 H) \u00b7 ~I\n~G \u2228 (~H \u2228 I) / G\n(A) Valid (B) Invalid. Counterexample when H and I are true and G is false (C) Invalid. Counterexample when H is true and I and G are false (D) Invalid. Counterexample when H, I, and G are false \nA: Let's think step by step.", "label": "(D)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\nConstruct a complete truth table for the following argument. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.)\n~(M \u2261 ~N)\nM / N\n(A) Valid (B) Invalid. Counterexample when M and N are true (C) Invalid. Counterexample when M is true and N is false (D) Invalid. Counterexample when N is true and M is false \nA: Let's think step by step.", "label": "(A)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic. Some CEO is wealthier than everyone. (Cx: x is a CEO; Px: x is a person; Wxy: x is wealthier than y)\n(A) (\u2203x)[Cx \u2022 (\u2200y)(Py \u2022 Wxy)] (B) (\u2203x)[Cx \u2283 (\u2200y)(Py \u2283 Wxy)] (C) (\u2203x)[Cx \u2022 (\u2200y)(Py \u2283 Wxy)] (D) (\u2200x)[Px \u2283 (\u2203y)(Cy \u2283 Wxy)] \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
{"question": "The following are multiple choice questions (with answers) about formal logic.\n\nQ: Which of the given formulas of PL is the best symbolization of the following sentence?\nTurtles live long lives and are happy creatures, unless they are injured.\n(A) (L \u2022 H) \u2261 I (B) (L \u2022 H) \u2228 I (C) L \u2022 (H \u2228 I) (D) L \u2022 (H \u2283 R).\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cL\u201d denote \u201cliving long\u201d, H \u201cbeing happy\u201d, and \u201cI\u201d \u201cbeing injured\u201d. Now, consider each choice:\n(A) means (living long AND being happy) is equivalent to (being injured). \n(B) means (living long AND being happy) OR (being injured). \n(C) means (living long) AND (being happy OR being injured). \n(D) means (living long) AND (being happy implies being R), but what R denotes is not clear.\nObviously, (B) is the best symbolization of the original sentence. The answer is (B).\n\nQ: Select the best translation into predicate logic.George borrows Hector's lawnmower. (g: George; h: Hector; l: Hector's lawnmower; Bxyx: x borrows y from z).\n(A) Blgh (B) Bhlg (C) Bglh (D) Bghl\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. We are told that \u201cBxyx\u201d means \u201cx borrows y from z\u201d. We can rewrite \u201cGeorge borrows Hector's lawnmower\u201d as \u201cGeorge borrows a lawnmower from Hector\u201d, which can then be translated into predicate logic as \u201cBglh\u201d. The answer \u201cBglh\u201d appears in (C); therefore, (C) must be the correct answer. The answer is (C).\n\nQ: \nSelect the best English interpretation of the given arguments in predicate logic.\nDm\n(\u2200x)(Wx \u2283 ~Dx). \n(\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax\n(A) Marina is a dancer. Some weaklings are not dancers. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball. (B) Marina is a dancer. No weakling is a dancer. Everything is either a weakling or plays volleyball. So something plays volleyball. (C) Marina is a dancer. Some weaklings are not dancers. Everything is either a weakling or plays volleyball. So something plays volleyball. (D) Marina is a dancer. No weakling is a dancer. Either everything is a weakling or Georgia plays volleyball. So something plays volleyball.\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cD\u201d denote \u201cbeing a dancer\u201d, \u201cm\u201d denote \u201cMaria\u201d, \u201cg\u201d denote \u201cGeorgia\u201d, \u201cW\u201d denote \u201cweakling\u201d, \u201cA\u201d denote \u201cplaying volleyball\u201d. Then, we have the following:\n1. Dm \u2192 Maria is a dance.\n2. (\u2200x)(Wx \u2283 ~Dx). \u2192 For all x, if x is a weakling, then x is not a dancer. In other words, no weakling is a dancer.\n3. (\u2200x)Wx \u2228 Ag\t/ (\u2203x)Ax \u2192 For all x, x is a weakling or Georgia plays volleyball. So there exists an x that plays volleyball. \nOptions (A) and (C) do claim that some weaklings are not dancers, but the second argument strongly states that no weakling is a dancer. Thus, we can eliminate them. Option (B) omits the important detail about Georgia playing volleyball. Option (D) has all the details presented in the arguments and is the best English interpretation of the arguments. The answer is (D).\n\nQ: Select the best translation into predicate logic: No people drive on Mars.\n(A) ~Pd (B) (\u2200x)(Px \u2228 ~Dx) (C) (\u2200x)(Px \u2283 ~Dx) (D) ~Dp\nA: Let's think step by step. We refer to Wikipedia articles on formal logic for help. Let\u2019s solve this step by step. Let \u201cP\u201d denote \u201cbeing on Mars\u201d and \u201cD\u201d denote \u201cdriving on Mars\u201d. Then let\u2019s consider each option:\nOption (A): ~Pd \u2192 d is not on Mars.\nOption (B): (\u2200x)(Px \u2228 ~Dx) \u2192 For all x, x is on Mars and x do not drive on Mars.\nOption (C): (\u2200x)(Px \u2283 ~Dx) \u2192 For all x, x is on Mars implies that x do not drive on Mars.\nOption (D): ~Dp: \u2192 p do not drive on Mars.\nOf all these options, Option (C) appears to be the best and most meaningful interpretation of the argument \u201cNo people drive on Mars.\u201d The answer is (C).\n\n Select the best translation into predicate logic. All children go to some school. (Cx: x is a child; Sx: x is a school; Gxy: x goes to y)\n(A) (\u2200x)(\u2203y)[(Cx \u2022 Sy) \u2022 Gxy) (B) (\u2200x)[Sx \u2283 (\u2203y)(Cy \u2022 Gxy)] (C) (\u2200x)[Cx \u2283 (\u2203y)(Sy \u2022 Gxy)] (D) (\u2200x)[Sx \u2022 (\u2203y)(Cy \u2022 Gxy)] \nA: Let's think step by step.", "label": "(C)", "options": ["(A)", "(B)", "(C)", "(D)"]}
